#128241

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂×C₈²
socle	C₂³	C₂×C₄×C₈
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6,8,11,9,7,10,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,30)(24,31)(25,29)(26,34)(27,32)(28,33)(35,41)(36,39)(37,40)(38,42)(43,44);(2,4)(13,15,16,23,17,24,26,35,18,25,27,36,28,37,38,43)(14,19,21,30,22,31,34,41,20,29,32,39,33,40,42,44);(5,7,8,12,9,6,10,11)(14,20)(19,29)(21,32)(22,33)(30,39)(31,40)(34,42)(41,44)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35437);